\(\mathcal{I}^{\mathcal{K}}\)-SEQUENTIAL TOPOLOGY

Abstract

In the literature, \(\mathcal{I}\)-convergence (or convergence in \(\mathcal{I}\)) was first introduced in [11]. Later related notions of \(\mathcal{I}\)-sequential topological space and \(\mathcal{I}^*\)-sequential topological space were introduced and studied. From the definitions it is clear that \(\mathcal{I}^*\)-sequential topological space is larger(finer) than \(\mathcal{I}\)-sequential topological space. This rises a question: is there any topology (different from discrete topology) on the topological space \(\mathcal{X}\) which is finer than \(\mathcal{I}^*\)-topological space? In this paper, we tried to find the answer to the question. We define \(\mathcal{I}^{\mathcal{K}}\)-sequential topology for any ideals \(\mathcal{I}\), \(\mathcal{K}\) and study main properties of it. First of all, some fundamental results about \(\mathcal{I}^{\mathcal{K}}\)-convergence of a sequence in a topological space \((\mathcal{X} ,\mathcal{T})\) are derived. After that, \(\mathcal{I}^{\mathcal{K}}\)-continuity and the subspace of the \(\mathcal{I}^{\mathcal{K}}\)-sequential topological space are investigated.

Keywords

• ideal convergence, \(\mathcal{i}^{\mathcal{k}}\)-convergence, sequential topology, \(\mathcal{i}^{\mathcal{k}}\)-sequential topology